Quote:
Originally Posted by nvman21
So the Eisenstein Criterion states that if f(x) is a non constant polynomaial of degree n with integer coefficents and there exists some prime p such that p divides every constant of f(x) but p^2 does not divide the constant term then f(x) is irreducible in Q[x]. I understand this but then in the examples it states:
x^4 + x^3 + x^2 +x +1 is irreducible because by the Eisenstein criterion (x+1)^4 + (x+1)^3 + (x+1)^2 + (x+1) +1= x^4 +5x^3 +10x^2 +10x +5 is irreducible. Somehow this is supposed to be obvious to the reader but I have no idea why the seconds function being irreducible implies that the first one is. Any thoughts or help?
If the latter polynomial had a root x, and thus was reducible, then the former polynomial would have a root (x-1), and thus would be reducible, and vice-versa.
EDIT: er, this doesn't cover it being reducible into quadratic, etc. factors, so there's a little more to it, don't remember off the top of my head
